PROBLEM 



ANALYSIS 



FORMULA 

INDETERMINATE 
SCIENCE-ANALYSIS 
DETERMINATE 



GEORGE ASHTON BLACK, PH.a 





,. J .J 'tjt.^ 



CopightN" 



i^i Q> 



CDFXRIGRT DEPOSIT. 



PROBLEM 

SCIENCE=ANALYSIS 



FORMULA 

INDETERMINATE 
SCIENCE=ANALYSIS 
DETERMINATE 

BY 
GEORGE ASHTON BLACK, Ph.D. 



Man gewinnt dadurcli schon sehr viel, ■wenn man eine 
Menge von UntersTichungen \inter die Formel einer 
einzigen Aufgabe bringen kann. Denn dadurcli erleich- 
tert man sich nicht allein selbst sein eigenes Geschaft, 
indem man es sicb genau bestimmt, sondem aueb jedem 
anderen, der es priifen will, das Urtheil, ob wir unserem 
Vorhaben ein Geniige gethan haben oder nicbt. — Kant. 



NEW YORK 

FEINTED FOR THE AUTHOR 

1916 



Copyright, 1916, by 
George Ashton BijACk 



Published September, 1916 



if ■ 

SEP 23 1916 

THE DE VINNE PRESS 



C!,A437787 






AXIOM 

If the analytical expression 

to be read equals^ is abstracted from any application of it, 
for instance in logic the classic application 

A=A 
and is considered by itself as a single principle of cognition, 
of some kind, then the mathematical construction 

a single principle f plane surface 

of cognition, of \ straight line — 

some kind [ two equal jiarallel straiglit lines = 

is found to determine only generally the singularity, in re- 
lation to the kind, of every single principle of cognition that 
was ever placed within the whole of the considered principle 
in the course of constructing it by degrees according to a 
constant rule; and this only general determination of a 
singular state of things is in turn found to prescribe the 
complete determination of the same. 

NOTE 

Increasing diversity in the application of a constant rule is 
the general but sufficient mark of deductive sequence in 
general. Invariably in this treatise 

successive = in the order of deduction successive 
fii'st=in the order of deduction first 

POSTULATE 

The mathematical construction 

{axiom negation 1 

postulate — reality \ quality- 
definition = limitation J 

can be recognized as necessary and sufficient indifferently 
either to make the complete determination prescribed in 
axiom a thing done, or to determine and resolve in deduc- 
tive sequence aU. cases of the singular general problem 

first cognition of any kind 

NOTE 

The same three principles of cognition in the same order, namely 

plane surface 
straight line — 

two equal parallel straight lines = 
3 



are also necessary and sufficient to construct mathemati- 
cally Kant's classic metaphysical theory whether of the dif- 
ferent successive steps in the process of making any object 
by degrees completely known according to a constant rule, 
or of what constitutes perfection in any cognition referred 
to any object. Witness 

{intuition synthesis 1 
concept — analysis [ method 
idea = dialectic J 

where the presence of two equivalent formulations of the 
theory is explained by the fact that Kant distinguishes the 
same cognitions, regarded from two different points of 
view, on the one hand as determinations of a known object, 
on the other hand as acts of a knowing subject. 

Use of classic analytical expressions in a new sense which 
can be gathered from the new context with mathematical 
certainty and precision has been made, not pointed out, 
hitherto; and will continue to be made, not pointed out, in 
the sequel. But in this place it is convenient to call atten- 
tion, once for all, to the use in question, by pointing to the 
example of it contained in the rest of this paragraph. The 
two equivalent formulations, when posited in one to one 
correspondence, but not yet mathematically constructed 

(intuition synthesis ] 
concept analysis - !■ method 

idea dialectic J 

perfectly differentiate the whole of possible cognition ac- 
cording to a constant rule, that is logically divide it into a 
series of empty compartments. The integration of the re- 
sulting differential equation, that is the filling of the said 
compartments by the only cognitions necessary and suf- 
ficient to fiJl them 

{intuition synthesis 1 
concept — analysis [method 
idea = dialectic J 

first reduces the formulated theory to practice; first con- 
structs it mathematically; first makes, according to that 
theory, the unknown cognitive function in the sense of 

4 



required cognition of any kind according to a constant rule, 
by degrees completely known. 

The construction of the two equivalent formulations of 
the theory raises a certain problem. For the resulting series 
in terms of mathematics 

plan^ surface 

straight line — 

two equal parallel straight lines = 

and the two equivalent formulations in terms of meta- 
physics 

{intuition synthesis 1 
concept analysis > method 

idea dialectic J 

differ as intuition and concept in respect of the same object, 
namely the theory. Therefore a corresponding idea remains 
to be invented or discovered. I find it in the necessary 
reference of the terms of the series to the comparison and 
distinction of them in respect of any singularity of any kind 
remarkable in any one of them. The resulting idea or 
dialectic 

negation 1 [If ^^^^ 

reality [ quality = -I — I = first ] postulate 
limitation J [ = J I definition 

supplies all that was wanting to the perfection of the first 
complete knowledge of the theory according to the theory 
itself as a constant rule. 
In the case of the notation 

intuition synthesis 
whether the indicated compartment is to be regarded as 
empty or as filled cannot be told on inspection of only that 
notation. If the two equivalent formulations of Kant's 
theory are merely posited in apposition to each other in a 
one to one correspondence constituting the differential 
equation 

{intuition synthesis 1 
concept analysis [method 

idea dialectic J 

then all the indicated compartments are empty and require 

5 



to be filled ; that is the theory requires to be reduced to 
practice ; that is the equation requires to be 
constructed = integrated = satisfied 

When all are filled by the only cognitions necessary and 
sufficient to fill them 

{intuition synthesis 1 
concept — analysis [method 
idea = dialectic J 

then the first compartment is in semblance still empty, but 
is in truth now filled by the cognition termed a plane in 
geometry. What is here in question is a certain new apph- 
cation of the classic mathematical principle of position ac- 
cording to which the same notation in different places has 
different meanings precisely determined by the context. For 
example, whether a straight hue is to be regarded as the 
analysis of a previously undivided plane into two equal parts 
that are connected in the line as a concept, or as the syn- 
thesis of the first part of the reproduction by degrees ac- 
cording to a constant rule of a plane equilateral triangle, of 
which part the hne is an intuition, depends in this treatise 
on the position of the Hne as immediately successive in the 
first case to the undivided plane, in the second case to the 
whole of the triangle. Another new application has already 
appeared in the case of the notation 

and stni another will presently appear in the case of the 

notation 

Science = Analysis 

In the most remarkable of these new applications the iden- 
tical notation in question is only a httle clear space some- 
how sufficiently indicated on the page. Two different mean- 
ings of it, as it appears when one and the same notation 
containing it 

intuition synthesis 

is met with in different places, have already been pointed 
out. Other different meanings of it in other different places 
remain to be pointed out in the sequel. 

6 



DEFINITION 

Science = cognition necessary and sufficient to re- 
solve all cases of a general problem in deductive 
sequence = Analysis. 

^ PROBLEM 

Science = Analysis 

FORMULA 

Indeterminate 
Science = Analysis 
Determinate 

WORK 

Science = Analysis 
Science | predicate | Analysis 

«^i-ee{-t7^,f^^^^^^^^^^ 

NOTE 

Formula is a convenient notation of the logical division 
necessary and sufficient to determine the case of problem 
universally through different cases in deductive sequence. 
The case in which problem is required to be indeterminate 
is the singular case. The case in which problem is required 
to be determinate is the general case. 

The first moment of work is the only solution of the 
singular case. In semblance it is only problem itself. In 
truth it is no determination of problem in reference to the 
necessary correlate of any subject of discussion, namely, 
any predicate. The second moment of work, as any deter- 
mination of problem in reference to any predicate, is the 
first of aU possible solutions of the general case. The third 
moment of work, as the sequence of negative and affirmative 
determination of problem in reference to any predicate, is 
the general solution to the form of which all solutions of the 
general case that are different and successive in reference to 
the first can be reduced. The sequence of second and 
third moments is necessary and sufficient to solve the 

7 



general case universally. The whole of work in giving the 
only solution of the singular case, and the only necessary 
and universal solution of the general case, gives the required 
resolution of all cases. 

PROBLEM 

Science | not any predicate | Analysis 

NOTE 
This negative problem is resolved only through the 
absence of formula and work in the case of it. According to 
definition it is not resolved through connecting with the 
word not the predicates resulting from the resolution of the 
next successive problem. 

PROBLEM 

Science | any predicate | Analysis 

FORMULA 

^-- { any sn'jl^sl^f ptdicate } ^^^^^ * 

WORK 

S-^--«{toc"on}^-^ly^i« 
according to classic definition of abstract in logic, of func- 
tion in mathematics. 

NOTE 
In logic classic distinction between abstract and applied 
so defines abstract that it cannot be any successive predicate, 
and can only be the first predicate, of which any successive 
predicate is some apphcation. In mathematics classic defini- 
tion, for instance Dirichlet's, requires function to be some 
determination of a variable in reference to a rule. As in 
any case either negatively or affirmatively determined in 
reference to a rule, the defined function cannot be the singular 
first predicate, andean only be the general any successive pred- 
icate, in which different successive predicates are connected. 



POSTULATE OF 
SCIENCE { ABSTRACT } ANALYSIS 
The solution of one case of Science { abstract } Analysis is 
first given or found in the shape of the single series 
o minimum maximum oo 

POSTULATE OF 

SCIENCE { FUNCTION } ANALYSIS 

The solution of one case of Science { function } Analysis is 

first given or found in the shape of the two connected series 

oo max. min. o min. max. c» 

NOTE 
One and the same degree in different places, for instance 
zero, has a certain singular character as proper to the single 
series, and a certain general character as common to the two 
connected series. In reference to the rule either plus or 
minus ± , every degree proper to the single series is indeter- 
minate, that is neither negatively nor af&rmatively deter- 
mined. In reference to the same rule any degree common 
to the two connected series is determinate, that is either 
negatively or affirmatively determined; in particular zero 
there as neither plus nor minus is negatively determined, 
and any other degree there as either plus or minus is affir- 
matively determined. The necessary and sufficient sign of 
zero as neither plus nor minus is only a little clear space in 
the case of zero corresponding to the space filled by the actual 
sign of either plus or minus in the case of any other degree. 
For example, in the whole of the notation 

o 

± CD 

the clear space in question is in truth a sign to be read 
neither plus nor minus. Comparison and distinction of 
degrees in Science { abstract } Analysis belong in Science 
{ abstract } Analysis. Comparison and distinction of degrees 
and signs in Science { function } Analysis belong in Science 
{ function } Analysis. 



Resolution of all cases of Science { abstract } Analysis achieved 
in and through the integration of a differential equation de- 
rived from the postulated solution of the first case. 



minimum 



maximum 



o 

CX) 

min. 
max. 



o 

00 

min. 
max. 



intuition 
concept 

idea 



ideal 



of Science | abstract | Analysis 



synthesis 1 
analysis I ^^^^^^ 

dialectic 



ui ^ 



minimum maximum 
o 



abstract ] 
mathematical 



00 



mm. 
max. 



l imi t 



minimum 1 ^i i^^ity 
maximum J * -' 



metaphysical { ^^^Zi::Z! } ^— ^^^^ 
logical! fj^ly"^^ I universality 



o 



gs 

Q ^ 
'-•' CO 



10 



Resolution of all eases of Science | function} Analysis acMeved 
in and through the integration of a differential equation de- 
rived from the postulated solution of the first case. 



00 max. 



mm. 



mm. max. 



o 

=1= 00 

^jmin. 
[max. 



o 

± 00 

^jmin. 
(.max. 



intiiitioii 
concept 

idea 



synthesis 
analysis 

dialectic 



ideal 



of Science | function | Analysis 



method 



P 00 



nun. 

min. 
max. 



o 

CO 



m 



van- 
^ able 



(s tMng 



miag. 
real 

no 
some 



o 
min. 
max. 

00 



individual 

general 

universal 

none 

one 

other 

one or other 



o 



function ^ g 



m 
I general 
any J 



immed. 



iindis. I thought 

[ concept 

distinct \ judgment \ mediate 

[ syllogism 

* reference of cognition to an object in general 



11 



NOTE 

Helpful to an understanding of the thing done on pages 10, 
11, is a new use of the circle comparable to the classic use 
of the same by Euler in logic. Witness 

80° O 



Ql 



o ^^ 

CD ^. 

§ § 

p 

CD 



r1- 
O 



c-K CD 



o 

pi 



a^ 


P 


s 


Pj 


CD 


P 


p 


CD 


g 


P^ 


CD 




^ 
^ 


t3" 


a 


CD 




i 

CD 


^ 


S 


CD 


cc 


CD 


^ 


13 


s 


•r$ 


CD 


i— ' 


i- 


PJ 



CD O 
C5 ■ 



CD ^ 

O 05 



CD CD 
CD ^ 



pj t^ Pj fcis 







CO o 



o 

B 
2. 







8 o 8 o 



cr5 cc 









2 ^ H 

00 ® O) 

'^'^ ^ 

^ (S M 

s * ® 

• o 2 
g S 

• ^ U 



12 



Also witness 



80O O 

H- H- 

IQI O 






^* *^ 

2 <^ 

r> 

P CO 

CO <J 

o p ^. 
a. p CD 

g: g ^ 

"^ ^ ^' 
5- fD o 

3' < ^ 
^ I— '• 

*<! !3. P 

t* ^ p 

p l-b 
p. g. 

p- g 5- 

s It 
^ B 

o S o 
I. M O 

® S § 

^ S'^ 



B 

a- 

*^ 
p^ 



2 ^ 

s ■ 



H- 



o o 

3 o 
a, a 

05 



oo p 2 



(I Et 20 

P -< 



P 



P' 

p" 



J:? £^. p5i 

g p rt- 
So® 

:i 






! P^ O 

cn5 P 



tf- r^ Hj (I) CO p ^ 



f^ ^. OQ P 

o p ^ "" 

^ g P'3 
o 



^ 1!=' S=! t" 



P.J3 



^3 



P o S' p 

o 1>'P P 

p ^ P P 

OQ I Pj Pj 



p-o 
® p 

PU 

3 ^ 

Vj. p. 

CD P. 



P5 


Th6 ¥el 

the who 
complet 
greesin 
ing the 




<r<- 


^ 


O 


4 trl- CD Sr P 




<I, B" !=-' <^ ?t- 


P. 

Si 


11^=8 


CD 


§ s |ta 


B 


^^ M S ^ 


p^ 


^ a§.-P^ 


CO 


Pj Pf P (S ® 

^ p ^g^s 




« 


2 S-.q- 5. ® 



o :^ 



S"^.B.If 



si^ 



;• o goo 



S- ^ 



P o r*» 

'SIS, 



O^ 



13 



In 6 it was a needed innovation to posit the classic and 
universal solution of the general case of 

tho'uglit= reference of cognition to an object in general 
in dependence upon the new found and only solution of the 
singular case. In 5 terms definitive of increasing diversity 
in metaphysical generality are compared and distinguished 
the identity and difference in which were hitherto imper- 
fectly discriminated. In and through 1, 2, 3, 4, is developed 
the doctrine or demonstrated theory of function through 
which and according to which any doctrine of different suc- 
cessive functions, for instance the formula 

Science | | Analysis 

Science | a? | Analysis 

and corresponding work 

Science | | Analysis 

(individual 1 
general > Analysis . 
universal J 

of developing the analytical expression 

Science | function | Analysis 

into a complete series, first becomes possible. 

The theory of functions in classic mathematics depends 
upon the two connected series of numbers 

oo . . . 3 (2 ^1 (0, 1, 2, 3 ... 00 

where zero is neither plus nor minus, and any other num- 
ber is either plus or minus. Implicit in the two connected 
series of signed numbers is the variety 

{ } 

1 1 

2, 3 ... [ =t 

CO J 

where the clear space, corresponding to the space filled by 
the actual sign of either plus or minus, is, in that corre- 
spondence, the necessary and sufficient sign of neither plus 
nor minus. The first comparison and distinction of the 

14 



numbers and signs conspicuous in the variety is recorded in 
the same notation as before in the case of t]ie corresponding 
variety of signed degrees on page 11. Therefore the notation 
expresses a common theory of function in respect of which 
classic mathematics and the present treatise are identical, 
however different they are in other respects. 
The notation in question 



variable 



imag. I } 

j individual 
real \ general 
universal 



function 



exhibits, in necessary and universal reference to function as 
any successive predicate, the sequence of one successive 
predicate conveniently read blank, and another suc- 

cessive predicate x, where the sequence of and x repre- 

sents any moment of the well-ordered logical determination 
of imaginary and real in necessary and universal reference to 
variable, apart from which, by definition, function is impos- 
sible. Since there is only one moment in the whole of the 
logical determination of imaginary, any moment of that 
determination is properly represented by the notation of 
that one in the place corresponding to the place of x repre- 
senting any moment of the logical determination of real. 



15 



POSTULATES OF 
SCIENCE { } ANALYSIS 

science! X } ANALYSIS 



I 

The solution of one case of Science { } Analysis is 

first given or found in the shape of a plane equilateral triangle 
the internal determination of which is only imaginary 



II 

The solution of one case of Science { individual } Analysis 
is first given or found in the shape of a plane equilateral 
triangle the internal determination of which is real in respect 
of only the middle point of the altitude 



m 



Resolution of all cases of Science {indi\T.dual} Analysis = 
solution of the first case of Science { general } Analysis. 



IV 



Resolution of aD. cases of Science {general} Analysis: 
solution of the first case of Science {universal} Analysis. 



HENCE 

by corresponding stages: 
16 



I Eesolution of all cases of Science { } Analysis achieved 
in and through the integration of a differential equation 
derived from the postulated solution of the first case. 



of 





* 1 








/ intuition 








/ concept 


ideal 






/\idea 












of Science 


} Analysis 


/ synthesis 








/ analysis 


method 






/ \ dialectic 










intuition 


A 


synthesis 






concept 




analysis 




ideal , 
Science 


idea /\ 


/ 
/ 
A 


dialectic 


, method 
of Analysis 



where the only imaginary predicate, as a part of speech, is 
an adjective which is read blank not only in two places 
picked out by small braces, but also in two corresponding 
places not so picked outj and where in 3 the clear space 
within the notation, concept analysis, and again at the 

right of the interior right hand brace, is a plane not first 
cognized in intuition as a plane pure and simple, but 
successively recognized in concept and in idea as the ground 
of the complete determination of the triangle as a whole and 
of the triangle as reproduced by degrees. 

17 



II Resolution, of aH cases of Science { individual } Analysis 
achieved in and through the integration of a differential 
equation derived from the postulated solution of the first case. 



ideal 
of individual 
Science 



intuition 

concept 

idea 

synthesis 

analysis 

dialectic 

intuition 



ideal 



of Science { individual } Analysis 



method 



synthesis 



concept ^ analysis 

A B 

AB 

X 

idea CD dialectic 

I! 

ABC-D 



method 

of individual 
Analysis 



18 



Ill Resolution of all cases of Science { general } Analysis 
achieved in and throngii the integration of a differential 
equation derived from the postulated solution of the first 
case, and sufBlciently distinguished as complex from prior 
differential equations as simple. 



C 

D 

A B 

AB 

X 

CD 

II 

ABC-D 



intuition 

concept 

idea 

synthesis 

analysis 

dialectic 



ideal 



of Science { general \ Analysis 



method 



synthesis 



method of general Analysis 

analysis dialectic 



ffi intuition 



eg concept 



Z 

D 
X Y 

multiplicand 



idea CD )4 altitude multiplier 



C 

D 

A B 

AB 

X 



base 

X 



II II 

ABC-D area 



product 



zZ 

D 

xX yY 

base = area -r- Yz alt. 

X XX 

i altitude = area -^ base 
11 II II 

area = area^-=- area 



where the complex form of the differential equation in 3 is 
rendered necessary by the specialty of the particular Science 
\ X \ Analysis in question. Only through the cross refer- 
ence of the two equivalent formulas obtained from 2 to the 
same whole of possible Science { general } Analysis, could it 
be perfectly differentiated a priori. 

19 



IV a Derivation from the postulated solution of the first 
case, of a compound differential equation demanding the 
resolution of all cases, of Science { universal } Analysis. 



A 

c 

D 

A B 

AB 



D 

X y 



Z 

D 

X Y 



zZ 

D 

xX yY 



base multiplicand 

XX X XXX 



base = area 4- }^ altitude 



CD }4 altitude multiplier }i altitude = area 4- base 
II II II II II II 

ABC'D area product area = area^4- area 



single 
definite 
individual 
2 



manifold 
definable 
general 



synthetical analytical 



restrictive intuition 

definitive concept > ideal 

universal idea I 

of Science | universal | Analysis 

dialectical synthesis 

" analysis [ method 

" dialectic 



3 Here belong the empty tables ABC which follow. 
They perfectly differentiate the whole of possible Science 
j universal } Analysis, and constitute a compound differen- 
tial equation demanding the resolution of all cases of that 
problem. 



20 



Pel 
o 
'A 

< 
>; 

fe 

o 
o 


> 














S 

o 

s 
















< 


syn. an. dia. synth. 


syn. an. dia. anal. 


syn. an. dia. dial. 


Me 


PHOD OF Universal Anali 
21 


'SIS 



s 

o 

i 
g 








> 












- 














syn. an. dia. synth.. 


syn. an. dia. 'anal. 


syn. an. dia. 


M 


ETHOD OF Universal Anai 
22 


.YSIS 



o 


» 






















Q 










syn. an. dia. synth. 


syn. an. dia. anal. 


syn. an. dia. dial. 


Mi 


:thod of Universal Anal 
23 


YSIS 



IV b Essay to refer recognized moments of the demanded 
resolution of all cases of Science { universal } Analysis each 
to its proper place in the formula demanding the resolution. 

On the supposition that pure reason, constant as the 
faculty of Science = Analysis in all rationals of all 
times, but varied through all moments of the more and 
more definite and particular use of that faculty according 
to the rule function in different rationals of different times, 
has somewhere in some context already cognized every step 
in the solution of every case of Science { universal } Analysis, 
but has not yet recognized any step in the solution of 
any case in its proper place in the resolution of all cases; 
I propose to search out all and only the cognitions that are 
the content of that resolution, and arrange them each in that 
proper place as fixed for it a priori by the empty tables ABC. 
To be sure the task is not for only one rational, but for 
every one interested in the development as much as 
possible in himself of the same faculty that aforetime made 
the cognitions, and is now in his person caUed upon to 
recognize what it has itself in other persons already cognized 
according to a fixed and ascertained formula. As my own 
discovery of the required cognitions and reference of them 
to this or that place in the formula is sure and complete as 
regards the solution of at least the first or singular case of 
the general problem in question, so all that is wanting to 
the perfection of the demanded resolution of all cases will 
undoubtedly be found, if able men, and such as are ac- 
quainted with what is classic in the use of pure reason, will 
endeavor to recognize the missing cognitions by the general 
bat sufficient marks that relegate them to one or another 
place in the formula in correlation with one or another mo- 
ment of the singular solution. 



24 



a 

o 

1 








% 






1 


















< 


<4^ 


< 




n >, >i '^ 

oQ NQ NO Nq 


AB X CD = ABCD 
base X Yz altitude = area 

multiplicand x multiplier = pr. 

base = area -=- % altitude 

yz altitude = area -=- base 
11 II II 
area = area2-j- area 


< 
EH 


syn. an. dia. synth. 


syn. an. dia. anal. 


syn. an. dia. dial. 










Method of Univeusal An 
25 


ALYSIS 







. — ' — , 


--H 


« 




III 


ill 






eS Art 


0,^3^ 


eS p^Ss 



1-c 



•S -3 -B P.fr 



§2S 



I 



111 §.1§ 




CO CO O "*;; cc "B 

pi cs » t^ 2 2 




S-E 



u_ 


ill 

.2 Mcrt 


5(-l 






^S 




is a-rd 


rt> 




Pi rt 


.2-2 


g 


111 


.2^ 


tPl 
II 




111 


1 II 


III 



O (D 



® ^ rt 



p! =s':3 

(0 eS'C 



fl as « 



Pl cS 



I Si 



m 

S ffi « 

rd P-Pl 



^ -s 



•-a 






■e S^ 
'So S 

2 rt ® 



® cS S 




sjTi. an. dia. synth. 




syn. 



an. dia. anal. 



S eS ® ® 

® pi's O 

,2 o'g o 

::3 ®«H A 



« ® 

li 

2 =* 

p CO 



hS 



C V 



syn. an. . dia. dial, 



Method of Universal Analysis 
26 



NOTE 
Table A in its present state is filled to only a certain extent, 
bnt so secures separate consideration of necessary and suf- 
ficient evidence that the distinctions comprehended under the 
title method of universal Analysis are not arbitrary, but 
represent corresponding differences in reality and truth. 
The same distinctions in the same order just as truly repre- 
sent corresponding differences in the groups that fill table 
B ; but the correspondence, unmistakable in table A, is dif- 
ficult, but not too difiicult, to detect in table B. 

At the end of table B, in three conspicuous compartments, 
is laid down what was hitherto lacking, a mathematically 
well ordered curriculum of the sciences deduced from a 
mathematical definition of science. The unit of the curric- 
ulum is one group. In reference to the first group as pro- 
totypal, the eleven different successive groups are ectypal. 
The first group is classic mathematics, meaning arithmetic 
algebra geometry once as separate sciences, and again as 
combined and raised to a higher power in the calculus. The 
organ on of the extension of our knowledge in respect of the 
first ectypal group is the present treatise. It deserves to 
be caUed posterior mathematics in reference to classic mathe- 
matics as prior. The next two groups it is convenient and 
exact to caU dynamics, in analogy with Kant's separation of 
the categories connected with relation and modahty as dy- 
namical, from the categories connected with quantity and 
quality as mathematical. It is also convenient and in con- 
formity with classic usage to call the first four groups phys- 
ics, the next four logic, the last four ethics. 

In prior mathematics there is no generally accepted notion 
of algebra corresponding to the notion of arithmetic as sci- 
ence of number. Nevertheless function is here required to 
be the subject-matter of algebra by the nature and position 
of the indicia discursive magnitude and plurality referred 
to quantity, corresponding to the derivation of function 
whether from signed number or from signed degree. 

27 



Aritlimetical number is usually termed positive, or posi- 
tive with exception of zero, which makes it at least generally 
congruent with a part of algebraical number. But accord- 
ing to the indicia discrete magnitude and unity referred 
to quantity, number in arithmetic cannot be congruent at 
all with number in algebra, and can only be, in contradistinc- 
tion to algebraical number, unsigned, absolute, abstract, as 
at least one mathematician, J. W. A. Young, already teaches. 
Corresponding to number unsigned in arithmetic and signed 
in algebra is space unsigned in synthetical geometry and 
signed in analytical geometry. 

In logic the sjrmbol of equivalence expresses a thought 
which as a concept is such that any different concept is 
subordinated to it according to the logical series 

predicate 
predicate{-*-y 

of which the logical equation 

predicate = predicate! ?-?y^^^ 

is only a transformation. The same thought expressed in 
the series and again in the equation is yet again expressed 
in the remark that the concept equals is the identical ground 
of the complete determination of any different concept. 
In logic also the state of logical extent, in case every degree 
thereof is neither negatively nor affirmatively determined 
in reference to the rule ±, is sufficiently indicated by the 
notation, log. ex, unsigned. The state of the same, in case 
any degree thereof is either negatively or affirmatively de- 
termined in reference to the same rule, is sufficiently indi- 
cated by the notation {log. ex. }± where the clear space 
to the left of the left-hand brace, corresponding to the space 
fiUed by the actual sign of either plus or minus, is, in that 
correspondence, the necessary and sufficient sign of neither 
plus nor minus. 

In ethics, according to the demonstration on pages 3, 4, 
of what constitutes perfection in any cognition referred to 

28 



any object, the production by degrees of the symbol of 
equivalence 



beginning with the inspection of the cognition termed a 
plane in geometry, first gives reason in the person of the 
producer something to do that it can do in respect of every 
one of its faculties in their proper order and connection. 
Accordingly such a production of this symbol is the neces- 
sary and sufficient means to the end of first educating the 
whole of reason in that person to a certain extent. First 
education is here definitive of any further education; for of 
course, in the order of deduction, only through first and 
according to first does any further ever become possible. 



29 



Deacidified using the Bookkeeper process. 
Neutralizing agent: Magnesium Oxide 
Treatment Date: Sept. 2004 

PreservatlonTechnologies 



